Existence of exotic 2-knots and orientable surfaces in S^4

Determine whether the 4-sphere S^4 contains any exotically knotted orientable surfaces—including 2-spheres (2-knots)—that are topologically isotopic to the standard embedding but not smoothly isotopic; equivalently, ascertain the existence or non-existence of exotic 2-knots in S^4.

Background

The paper constructs exotic 2-links in closed simply-connected 4-manifolds for arbitrary finitely presented groups and discusses conditions under which these links can be made nullhomotopic. Despite these advances in larger ambient manifolds, the authors emphasize that the fundamental question for the 4-sphere S^4 itself remains unresolved.

They note that analogous situations are settled in bigger simply-connected 4-manifolds (e.g., nullhomologous exotic 2-knots or tori), but the existence of exotic orientable surfaces in S^4—particularly exotic 2-knots—has not been established. This highlights a central open problem at the interface of smooth and topological 4-dimensional knot theory.